Optimal block designs for diallel crosses

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DOI 10.1007/s00184-009-0236-5

Optimal block designs for diallel crosses

M. K. Sharma · Sileshi Fanta

Received: 9 June 2008 / Published online: 26 February 2009

© The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract Dey and Midha (Biometrika 83(2):484–489, 1996) constructed optimal block designs for complete diallel cross experiment using triangular partially bal-anced incomplete block designs with two associate classes. They listed optimal block designs for the lines in the range from 5 ≤ v ≤ 10. In this paper, we are also pro-posing additional optimal block designs for complete diallel cross experiment using two associate class partially balanced block designs for some additional values ofv. Our method yields designs for proper and non-proper settings for complete diallel cross experiments. The proper and non proper designs are optimal in the sense of Kempthorne (Genetics 41:451–459, 1956) and non-proper designs are universally optimal in the sense of Kiefer (A survey of statistical design and linear models, North Holland, Amsterdam, 1975). The list of practically important designs is also given. Keywords Partially balanced incomplete block design· Complete diallel cross · General combining ability· Mating–environment design · Auxiliary design · Efficiency

1 Introduction

A diallel cross consists of all possible crosses between a numbers of varieties. Recip-rocal crosses and the selfed parents may or may not be omitted. Diallel crosses as a mating design is used to study the genetic properties of inbred lines in plant breeding experiments.

M. K. Sharma (

B

)· S. Fanta

Department of Statistics, Addis Ababa University, P.B. No. 1176, Addis Ababa, Ethiopia

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Among the four types of diallels discussed byGriffing(1956), system IV is the most commonly used in plant breeding. Suppose there arev lines and let us consider a cross of the form i× j with i < j = 1, . . . , v. With all possible nc = v(v − 1)/2

crosses. This is sometimes referred to as the modified or half-diallel. We shall refer to it as a complete diallel cross (CDC).

The common practice with diallel cross experiment is to evaluate the crosses in completely randomized designs or randomized complete block designs as environ-ment designs, e.g.Kempthorne and Curnow(1961). Due to limitation of homogeneous experimental units in a block to accommodate all the chosen crosses, the estimate of genetic parameters would not be precise enough if a complete block design was adopted for large number of crosses. To overcome this problem, many researchers used bal-anced incomplete block (BIB) designs, partially balbal-anced incomplete block (PBIB) designs with two associate classes etc. by treating the crosses as treatments. These designs have interesting optimality properties when making inferences on a complete set of orthonormalised treatment contrasts. However, in diallel cross experiments the interest of the experimenter is in making comparisons among general combining abil-ity (gca) effects of lines and not crosses and therefore, using these designs as mating designs may result into poor precision of the comparison among lines. Further, the analysis of a diallel cross experiment in incomplete block depends on the incidence of lines rather than the incidence of the crosses as treatments with in a block. It is there-fore apparent that special techniques are required to obtain good designs for diallel crosses experiments.

Several authors such as Gupta and Kageyama(1994), Dey and Midha (1996),

Mukerjee(1997),Das et al.(1998),Parsad et al.(1999) andSharma(2004) addressed the problem of finding optimal designs by using nested incomplete block designs (NBIB), triangular PBIB designs, nested balanced block (NBB) designs, GD PBIB designs and circular designs, etc.Gupta and Kageyama(1994) andSharma(2004) reported optimal designs in which every cross is replicated once but their designs dif-fer in their parametric values with proposed designs.Das et al.(1998) andParsad et al.

(1999) reported optimal designs for single as well more replications.Dey and Midha

(1996) reported optimal and efficient designs in which the crosses are replicated in the range 3≤ r ≤ 10. These designs also differ in parametric values of our proposed designs.

We, in this paper, derive additional incomplete block designs for the same mat-ing designs, usmat-ing two associate class PBIB designs such that none of the λ’s is zero. We have also listed these designs for reasonable practically usable values along with designs reported by these authors. The model considered involves only the gca effects .The specific combining ability (sca) effects being excluded from the model because the derived designs are not connected for cross effects. The paper is structured as: (1) the method of construction of designs is presented in Sect. 2, (2) in Sect.3analysis and optimality of the designs is considered and we show that the non-proper designs have strong optimality properties, (3) in Sect. 4, the effi-ciency factor of both designs (proper and non-proper) as compared to randomized block designs is considered. For definition and properties of PBIB design, seeDey

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2 Method of construction

The method is simply stated as: from tables ofClatworthy(1973), take forv lines under evaluation and numbered randomly, a two associate PBIB design with param-etersv = b, r = k, λ1, λ2, n1, n2, pij k (i, j, k = 1, 2) having none of the λ’s equal to zero and also with the property that any pair of treatments does not occur more than once in any column of the design when we consider block size k as a blocks. We call this design as an auxiliary design. This feature of the auxiliary design helps us to construct a block design for CDC.

Now in auxiliary design, take all possible distinct pair of combinations of treatments in each block, starting from the first treatment of the block. These give k(k − 1)/2 pairs per block. Thus we get resulting design in which bk(k − 1)/2 pairs arranged in

b blocks, each containing k(k − 1)/2 pairs. Now in resulting block design identify

these pairs of treatments as crosses by treating treatments of the original design as lines. Now we consider in resulting design, the number of plots(= k(k − 1)/2) as blocks and number of blocks(b = v) as the block size of each k(k − 1)/2 blocks. We call this arrangement as mating design and denote it as d. Sinceλ’s are unequal, it leads to unequal repetition of the crosses in design d. The total number of crosses in mating design d, is bk(k − 1)/2 and out of these vni/2 crosses are appearing in

niλi/2 blocks (i = 1, 2). Since replications of crosses are unequal, it makes mating

design unbalanced for CDC experiment. To make the design balanced for CDC exper-iment, we will have to make the replications of the crosses equal because in balanced designs each elementary contrast among gca effects is estimated with equal precision under the assumptions of homogeneous error variance across all blocks. To do this, if λ1> λ2we delete 12[n1(λ1− λ2)] blocks, (or, if λ2> λ1then12[n2(λ2− λ1)]) from

k(k − 1)/2 blocks in design d.

In some auxiliary designs, forv even lines, if (λ1−λ2) (or (λ2−λ1)) is odd, then n1 (or n2) is also odd, then the expression of number of blocks to be deleted, will not be a positive integer but it will be equal to some positive integer±0.5. So in this case we will have to delete blocks equal to some value of positive integer and 0.5 fraction of one of the block which contains repeated crosses i.e. number of crosses to be deleted are equal tov× (value of the integer ± v/2). The process of deletion of blocks will be done with the help of association schemes of auxiliary designs (i.e PBIB designs). Now we may classify our auxiliary designs into two classes (1) where [n1(λ1− λ2)] and [n2(λ2− λ1)] are both positive even integers and (2) where [n1(λ1− λ2)] and [n2(λ2− λ1)] are both odd positive integers. We denote both these auxiliary designs as d(1)and d(2).

Thus the process of deletion of blocks of repeated crosses will yield two types of mating–environment designs (proper and non-proper) for CDC experiments with parameters.

(i) v1= v(v − 1)/2, b1= λ2(v − 1) if λ1> λ2(orλ1(v − 1) if λ2> λ1), r1= λ2 (orλ1), k1= v

(ii) v2= v(v − 1)/2, b2= λ2(v − 1) if λ1> λ2(orλ1(v − 1) if λ2> λ1), r2= λ2 (orλ1), k2= (v, v/2)

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Plan (auxiliary design d(1)) Treatment First associate Second associate B1 1 2 4 1 2, 5 3, 4 B2 2 3 5 2 1, 3 4, 5 B3 3 4 1 3 2, 4 1, 5 B4 4 5 2 4 3, 5 1, 2 B5 5 1 3 5 1, 4 2, 3

Design d Design d1(mating–environment design)

B1 B2 B3 B1 B2 1× 2 1× 4 2× 4 1× 2 1× 4 2× 3 2× 5 3× 5 2× 3 2× 5 3× 4 3× 1 4× 1 3× 4 3× 1 4× 5 4× 2 5× 2 4× 5 4× 2 5× 1 5× 3 1× 3 5× 1 5× 3

Plan (auxiliary design d(2)) Treatment First associate Second associate

B1 1 2 4 1 4 2, 3, 5, 6 B2 2 3 5 2 5 1, 3, 4, 6 B3 3 4 6 3 6 1, 2, 4, 5 B4 4 5 1 4 1 2, 3, 5, 6 B5 5 6 2 5 2 1, 3, 4, 6 B6 6 1 3 6 3 1, 2, 4, 6

Design d Design d2(mating–environment design)

B1 B2 B3 B1 B2 B3 1× 2 1× 4 2× 4 1× 2 1× 4 2× 4 2× 3 2× 5 3× 5 2× 3 2× 5 3× 5 3× 4 3× 6 4× 6 3× 4 3× 6 4× 6 4× 5 4× 1 5× 1 4× 5 5× 1 5× 6 5× 2 6× 2 5× 6 6× 2 6× 1 6× 3 1× 3 6× 1 1× 3

We denote both designs as d1 and d2. The method of construction is illustrated below by two examples.

Example 1 For illustration, we consider design C12 (Clatworthy 1973) with param-eters v = b = 5, r = k = 3, n1 = n2 = 2, λ1 = 1 and λ2 = 2. The plan and association scheme of the design is given below:

Sinceλ2= 2, the crosses (1 × 3), (1 × 4), (2 × 4), (2 × 5) and (3 × 5) appeared in both blocks 2 and 3 of design d. So we will delete one of the blocks to obtain M–E design d1.

Example 2 For second type of design, we consider design R 42 (Clatworthy 1973) with parametersv = 6, r = 3, k = 3, b = 6, m = 3, n = 2, λ1= 2 and λ2= 1.

Sinceλ1= 2, the crosses (1×4), (2×5), and (3×6) appeared repeatedly in block 2 of design d. So we will delete these crosses from block 2 to keep the replication same for all crosses. Thus we obtain d2.

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3 Analysis and optimality

For convenience in further discussion, we will denote both designs d1and d2as d∗.The data obtained from the design d∗, we take the following model:

Y= µ1n+ 1g+ 2β + e (3.1)

where Y is the n× 1 observational vector, µ is the general mean, 1n denotes an

n-component vector of 1’s, g=(g1, . . . , gv)andβ = (β1, . . . , βb) are the vector of v

gca effects and b(= b1 = b2) block effects respectively. 1and2are the corre-sponding design matrices of of n× v and n × b, respectively; that is (s, i)th element of1is 1 if the cross in the sth experimental unit has one parent i and is 0 otherwise. Similarly(s, u)th element of 2is 1 if the cross in the sth experimental unit comes from uth block and 0 otherwise. e is a random vector of error components and takes care of specific combining ability as well as unassignable variation and distributed with mean 0 and constant varianceσ2.

For 1≤ i < j ≤ v, let gdi jis the number of times cross(i × j) occurs in d∗. Let

sdi be the number of times the i th line occurs in design d∗.

FollowingGupta and Kageyama(1994), it can be shown that the information matrix for g under d∗is

Cd= Gd− v−1Nd∗Nd∗ (3.2)

where11= Gd= (gdi j), gdii = sdi and12= Nd= (ndi j) is the v × b

matrix of parental lines versus blocks, ndi j is the number of times line i occurs in

block u.

Under the model, the reduced normal equations for gca, using design d∗, are

Cd∗gˆ= Q (3.3)

where Q= T−v−1NdB. Here T is the vector of line total and B is the vector of block

totals. FollowingDey and Midha(1996) we now have the following results which will help in obtaining Cd-matrix for both types designs i.e. d∗.

Lemma 3.1 For the design d, the following are true.

(i) b  u=1 nil = λ1(n1+ n2) = λ1(v − 1) if λ2> λ1 = λ2(n1+ n2) = λ2(v − 1) if λ1> λ2 v  i=1 nil = 2v

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(ii) b  u=1 nil2 = 2λ1(n1+ n2) = 2λ1(v − 1) if λ2> λ1 = 2λ2(n1+ n2) = 2λ2(v − 1) if λ1> λ2 (iii) b  u=1 nilnil = 2λ1(n1+ n2) = 2λ1(v − 1) if λ2> λ1 = 2λ2(n1+ n2) = 2λ2(v − 1) if λ1> λ2

The proofs of all identities are easy are omitted.

From Lemma (3.1), it follows that, for the design d, Cd∗is given by

Cd= θ



Iv− v−11v1v 

(3.4) whereθ = λ1(v − 2) or λ2(v − 2), Ivis the identity matrix of orderv and 1vis thev component vector of all 1’s.

From (3.4) it is easy to see that generalized inverse of Cd∗is.

Cd= θ−1Iv (3.5)

It is obvious from (3.5) that d∗is a variance balanced and therefore all elementary contrasts among gca effects are estimated under the assumption of homogeneous error variance across all blocks with a variance 2σ2/θ. Hence design dis connected and has rank equal tov − 1. Since design d(2)has blocks of unequal block sizes, therefore the error variance of d(2)will not be the same as of design d(1). Letσ12andσ22be the error variances of designs d(1)and d(2), respectively. Now we can interpret that all elementary contrasts among gca effects in designs d(1)and d(2)are estimated with a variances 2σ12

θ and

2σ2 2

θ , respectively. Further the adjusted sum of squares due to

gca effects is simplyθ−1QQ= θ−1(Q21+ . . . +Q2v), where i = 1, 2 . . . v, Qi is the

adjusted total of the i th line; that is Qi is the i th component of the vector Q, defined

in (3.3). We thus have the following result.

Theorem 3.1 The design d(i.e. d1and d2) is variance—balanced for general com-bining ability effects.

Now we take the optimality aspects. The optimality criterion is the minimization of average variance of the best linear unbiased estimators of all elementary comparisons between gca effects.

Let D(v, b, k1, . . ., kb) denote the class of all connected block designs d∗withv

lines, b blocks such that j th block is of size kj. Similarly D0(v, b, n) denote the class

of all connected block designs d∗withv lines, b blocks and n experimental units. Here the block sizes are arbitrary but for a given design dεD0(v, b, n), the block sizes are equal.

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Theorem 3.2 Let d∈ D(v, b, k1, . . ., kb)[D0(v, b, n)] be a block design for diallel

crosses and suppose that dsatisfies

(i) Trace(Cd) = 2

b j=1kj

b

j=1k1j[2kj(2xj+ 1) − vxj(xj+ 1)] if and only

if ndi j = xjor xj+ 1 for all i = 1, . . ., v; j = 1, . . ., b, where xjis a positive

integer. And for ndi j = 0 or 1, this reduces to [Trace (Cd) = 2(n −b)], where

n and b are total number of experimental units and number of blocks in diallel

cross design [D0(v, b, n)], respectively. (ii) Cd∗is completely symmetric.

Then dis universally optimal over D(v, b, k1, . . ., kb)[D0(v, b, n)]

For any member of D(v, b,k1, . . ., kb) [D0(v, b, n)], we have Trace (Cd) = λ2(v−

1)(v −2) if λ1> λ2orλ1(v −1)(v −2) if λ2> λ1and for any member D0(v, b, n) to be universally optimal, the Trace(Cd) must be equal to 2(n − b), where 2(n − b) =

λ2(v − 1)2ifλ1> λ2(orλ1(v − 1)2ifλ2> λ1). The equality of Trace (C

d) follows

for any member of D(v, b, k1, . . ., kb) but for any member of [D0(v, b, n)], the Trace

(Cd) is less than 2(n − b).

Hence we have the following theorem.

Theorem 3.3 The designs d2obtained from d(2),a two associate PBIB designs, where none of theλ’s is zero, are universally optimal over D(v, b,k1, . . ., kb).

4 Efficiency factor

Now we will show that the designs d(1)and d(2)are optimal in the sense ofKempthorne

(1956). If instead of the designs d(1)and d(2), one adopts a randomized complete block design with r1(r2) blocks, each block having all the v(v − 1)/2 crosses , the C matrix of the randomized block design i.e. CR-matrix can easily shown to be, seeDey and Midha(1996).

CR= ri(v − 2)(Iv− v−11v1v) where i = 1 or 2 (4.1)

Hence the variance of the best linear unbiased estimator of any elementary contrast among gca effects in the case of randomized block experiment is 2σ2/r

i(v−2), where

σ2is the per observation variance. Thus the efficiency factors e1and e2, respectively of the design d(1)and d(2), relative to randomized complete block designs under the assumption of equal intrablock variances is given by

e1= θ/r1(v − 2) = r1(v − 2)/r1(v − 2) = 1 (4.2) and

e2= r2(v − 2)/r2(v − 2) = 1 (4.3)

whereθ = r1(v − 2) (or r2(v − 2)) for design d(1)(or d(2)).

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Table 1

Sl. no. Ref. no. No. of No. of No. of No of Total no. of Design names of the lines blocks crosses replication experimental reported by

design to be to be of the units different

deleted deleted crosses (r ) authors

1 S2 6 1 3 2 30 F2,Parsad et al. (1999) andDas et al.(1998, Th. 4.1) 2 S19 8 1 4 4 112 N 3 S23 9 3 9 3 108 F2,Parsad et al. (1999) 4 S29 12 4 12 2 132 N 5 S33 14 2 7 2 182 N

6 S42 21 5 21 1 210 Ser. 1,Gupta and

Kageyama(1994)

andSharma

(2004)

7a S44 26 2.5 13 1 325 Ser. 2,Gupta and

Kageyama(1994) andSharma (2004) 8 S52 10 1 5 6 270 N 9 S60 14 2 7 4 364 N 10 S68 20 9 30 2 380 N 11 S72 26 3 13 2 650 N 12 S90 21 6 21 3 630 F2,Parsad et al. (1999) 13 S93 30 7 30 2 870 N 14 S99 12 1 6 8 528 N 15 S104 15 10 30 5 525 F2,Parsad et al. (1999) 16a S105 18 2.5 9 5 765 F2,Parsad et al. (1999) 17 S111 22 3 11 4 924 F2,Parsad et al. (1999) 18 S115 30 3 60 2 870 N 19 SR9 9 3 27 3 108 F2,Parsad et al. (1999) 20 SR68 12 4 48 2 132 N

21a R42 6 0.5 3 1 15 Ser.2,Gupta and

Kageyama(1994) andSharma (2004) 22 R94 6 1 6 2 30 F2,Parsad et al. (1999) 23 R104 9 2 9 1 36 F1,Parsad et al. (1999) and F5, Das et al.(1998)

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Table 1 continued

Sl. no. Ref. no. No. of No. of No. of No of Total no. of Design names of the lines blocks crosses replication experimental reported by

design to be to be of the units different

deleted deleted crosses (r ) authors

24a R109 12 0.5 6 1 66 Ser. 2,Gupta and

Kageyama(1994) andSharma (2004) 25 R133 8 3 12 2 56 N 26 R134 8 3 24 2 56 N 27 R137 9 2 9 2 72 F4,Das et al. (1998) 28 R139 10 1 5 2 90 F2,Parsad et al. (1999)

29a R145 12 4.5 54 1 66 Ser. 2,Gupta and

Kageyama(1994) andSharma (2004) 30 R166 10 6 20 2 90 F2,Parsad et al. (1999) 31 R168 15 8 30 1 105 F1,Gupta and Kageyama(1994) andSharma (2004) 32 R170 27 2 27 1 351 F1,Parsad et al. (1999) and F5, Das et al.(1998)

33a R171 28 1.5 42 1 378 Ser. 2,Gupta and

Kageyama(1994) andSharma (2004) 34 R172 9 1 9 5 180 F2,Parsad et al. (1999) 35 R173 12 10 30 5 330 F2,Parsad et al. (1999) 36a R174 12 4.5 54 3 198 F2,Parsad et al. (1999) 37 R175 12 10 30 2 132 F2,Parsad et al. (1999) 38a R176 12 4.5 54 3 198 N 39a R177 14 1.5 7 3 273 N

40a R178 18 12.5 45 1 153 Ser. 2,Gupta and

Kageyama(1994) andSharma (2004) 41 R179 20 2 40 2 380 N 42 R180 20 2 10 2 380 N 43a R186 12 0.5 6 5 330 F2,Parsad et al. (1999)

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Table 1 continued

Sl. no. Ref. no. No. of No. of No. of No of Total no. of Design names of the lines blocks crosses replication experimental reported by

design to be to be of the units different

deleted deleted crosses (r ) authors

44 R187 14 15 42 2 210 N

45 R188 21 18 63 1 210 Ser. 1,Gupta and

Kageyama(1994) andSharma (2004) 46 R189 24 5 60 2 552 N 47 R193 12 3 18 6 396 N 48 R194 15 8 30 4 420 F2,Parsad et al. (1999) 49 R195 16 21 56 2 240 F2,Parsad et al. (1999) 50 R196 18 2 9 4 612 N 51 R196 18 2 9 4 612 N

52a R198 24 24.5 84 1 276 Ser. 2,Gupta and

Kageyama(1994) andSharma (2004) 53 R200 28 9 84 2 756 F2,Parsad et al. (1999) 54 R204 14 6 42 6 546 F2,Parsad et al. (1999) 55 R205 14 6 84 6 546 F2,Parsad et al. (1999) 56 R206 18 28 81 2 756 F2,Parsad et al. (1999) 57 R207 27 32 108 1 351 F1,Parsad et al. (1999) and F5, Das et al.(1998)

58a T33 10 1.5 15 1 45 Ser. 2,Gupta and

Kageyama(1994) andSharma (2004) 59 T58 10 3 45 2 90 F2,Parsad et al. (1999) 60a T60 10 1.5 15 3 135 N

61 T61 15 8 60 1 105 Ser. 1,Gupta and

Kageyama(1994) andSharma (2004) 62 T71 10 3 15 4 180 N 63 T84 15 8 60 4 420 F2,Parsad et al. (1999) 64 T94 21 15 105 3 630 F2,Parsad et al. (1999)

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Table 1 continued

Sl. no. Ref. no. No. of No. of No. of No of Total no. of Design names of the lines blocks crosses replication experimental reported by

design to be to be of the units different

deleted deleted crosses (r ) authors

65 T95 21 5 105 4 820 N 66 LS26 9 2 18 1 36 F1,Parsad et al. (1999) and F5, Das et al.(1998) 67 LS49 9 2 18 2 72 F4,Das et al. (1998) 68 LS72 9 2 18 2 72 F4,Das et al. (1998) 69 LS83 16 6 48 2 240 F2,Parsad et al. (1999) 70 LS99 16 9 72 3 360 F2,Parsad et al. (1999) 71 LS100 16 6 48 3 360 F2,Parsad et al. (1999) 72 LS101 25 4 100 2 600 N 73 LS116 16 6 48 4 480 N 74 LS117 25 12 100 2 600 N 75 C12 5 1 5 1 10 F1,Parsad et al. (1999) and F5, Das et al.(1998) 76 C23 13 3 39 2 156 N 77 C24 13 3 39 3 234 N 78 C25 29 7 203 1 406 F1,Parsad et al. (1999) and F4, Das et al.(1998) 79 C26 17 4 68 3 408 F2,Parsad et al. (1999) 80 C27 29 7 203 1 406 F1,Parsad et al. (1999) and F5, Das et al.(1998) 81 C28 17 4 68 4 544 N 82 C29 13 3 39 7 468 F2,Parsad et al. (1999) 83 M34 16 6 48 4 480 N 84a M38 26 7.5 195 3 975 F2,Parsad et al. (1999) 85 M39 27 32 216 1 351 F1,Parsad et al. (1999) and F5, Das et al.(1998)

N new, Ser. series, F family

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5 Conclusion

Optimal block designs with proper and non-proper settings have been proposed for CDC system IV using some PBIB designs. The non-proper setting designs found to be universally optimal in the sense ofKiefer(1975) and proper and non-proper setting designs are optimal in the sense ofKempthorne(1956). These designs retain full effi-ciency for the estimation of the contrast of interest. We investigated 85 PBIB designs (Clatworthy 1973). Out of which 20 and 29 PBIB designs gave block designs for CDC experiment in which each cross is replicated once and twice, respectively and in rest of the designs each cross is replicated more than twice. These designs are in the range of 5≤ v ≤ 30, except for v = 11, 17, 19, 22, and 23.

Acknowledgments The authors wish to thank the Editor and both the referees for their valuable comments and suggestions that greatly improved the presentation of the paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution Noncom-mercial License which permits any noncomNoncom-mercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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